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The Principal Principle, as Lewis stated it, includes an admissibility condition. Any adequate account of admissibility should entail Conditions 1 and 2 (see below). Together with Conditions 1 and 2, the Principal Principle entails the Principle of Indifference. Thus, the Principal Principle entails the Principle of Indifference.

Read like this, my response to the argument ran thus:

There is an account of admissibility -- namely, Levi-admissibility -- that is adequate and on which Condition 2 is not generally true. Levi-admissibility is adequate since has all of the features that Lewis required of admissibility, and it is very natural when we consider a close relative of Lewis' Principal Principle, namely, Levi's Principal Principle, which follows from Lewis' Principal Principle given some natural assumptions about admissibility that Lewis accepts.

However, there is another reading of the HLWW argument, and indeed it seems that some of H, L, W, and W favour it. On this alternative reading, it is not assumed that Conditions 1 and 2 follow from any adequate account of admissibility. Rather Conditions 1 and 2 are not taken to be consequences of the Principal Principle at all. Rather, they are intended to be plausible further constraints on credences that are independent of the Principal Principle. Thus, on this reading, the conclusion of the HLWW is not that the Principal Principle implies the Principle of Indifference. Rather, it is that the Principal Principle, together with two further norms (namely, Conditions 1 and 2), implies the Principle of Indifference.

In this post, I will raise an objection to this alternative argument.

The HLWW argument turns on a mathematical theorem. It takes certain constraints -- (I), (II), (III) below -- and shows that, if an agent's credence function satisfies those constraints, then it must satisfy a particular instance of the Principle of Indifference.

Thus, to obtain this from Theorem 1, we need the following: for each atomic $F$, there is $A$, $X$, and $0 < x < 1$ that satisfy (I), (II), and (III). Conditions 1 and 2 are intended to obtain this, but I think the argument is clearest if we argue for them directly, using the considerations found in HLWW.

Thus, suppose $F$ is atomic. Then the idea is this. Pick a proposition $X$ with two features: (a) if you were to learn $X$ and nothing more as your first piece of evidence, it would place a very strict constraint on your credence in $A$ --- it would require you to have credence $x$ in $A$; (b) $X$ provides no information about $F$ nor about the relationship between $A$ and $F$. Now, providing that $A$ is not logically related to $F$, we might take $X$ to be the proposition $C^A_x$ that says that the objective chance of $A$ is $x$. By the Principal Principle, $C^A_x$ has the first feature (a): $P_0(A | X) = x$. What's more, since $A$ is logically independent of $F$, $C^A_x$ also has the second feature (b): in the absence of further evidence, and in particular evidence about the relationship between $A$ and $F$, $C^A_x$ provides no information about $F$ nor about the relationship between $A$ and $F$.

Now, with $A$, $X$, $x$ in hand, we appeal to two principles concerning the way that we should respond to evidence:

(Ev1): If your credence function is $P$ and your evidence does not provide any information about the connection between $B$ and $C$, then $P(B | C) = P(B)$.

In slogan form, as we will see: Credences supported by stronger evidence are more resilient.

Now, from (Ev1), we immediately obtain (I) for our agent's initial credence function $P_0$ with $F$ atomic and $X = C^A_x$. After all, if you have no evidence, your evidence certainly does not provide any information about the connection between $C^A_x$ and $F$.

Thus, the plausibility of the HLWW argument turns on the plausibility of (Ev1) and (Ev2). Unfortunately, both beg the question concerning the Principle of Indifference. As a result, they cannot be assumed in a justification of that norm. Let's consider each in turn.

First, (Ev1). If your evidence does not provide any information about the connection between $B$ and $C$, then this evidence leaves open the possibility that $B$ is positively relevant to $C$; it leaves open the possibility that $B$ is negatively relevant to $C$; and it leaves open the possibility that $B$ is irrelevant to $C$. But (Ev1) demands that we deny the first two possibilities and take $B$ to be irrelevant to $C$. But why? Without further argument, it seems that we would be equally justified in taking $B$ to be positively relevant to $C$ and equally justified in taking $C$ to be negatively relevant to $C$.

Second, (Ev2). The idea is this: When I learn that two propositions, $B$ and $C$, are equivalent, there are many ways I might respond. I might retain my prior credence in $B$ and bring my credence in $C$ into line with that. Or I might retain my prior credence in $C$ and bring my credence in $B$ into line with that. Or I might do many other things. (Ev2) says that, if I have strong evidence concerning $B$ and no evidence concerning $C$, then I should opt for the first response and retain my prior credence in $B$ -- which was formed in response to the strong evidence concerning $B$ -- and bring my credence in $C$ into line with that -- since my prior credence in $C$ was, in any case, formed in response to no relevant evidence at all.

Now, on the face of it, this seems like a reasonable constraint on our response to evidence. It says, essentially, that credence formed in response to stronger evidence should be more resilient than credence formed in response to weaker evidence. And, as a limiting case, credence formed in response to strong evidence, such as evidence about the chances, should be maximally resilient when compared to credence formed in response to no evidence. (Note that a similar way of thinking might give an alternative motivation for (II), since this is also a principle of resilient credence.)

However, unfortunately, (Ev2) threatens to be inconsistent. After all, it is easy to suppose that there are propositions $B$, $C$, and $D$ such that you have strong evidence for $B$, but no evidence concerning $C$ or $D$ or $C\ \&\ D$ or $C\ \&\ \neg D$. But, in that situation, (Ev2) entails:

$P(B | B \leftrightarrow C) = P(B)$

$P(B | B \leftrightarrow (C\ \&\ D)) = P(B)$

$P(B | B \leftrightarrow (C\ \&\ \neg D)) = P(B)$

And unfortunately these are inconsistent constraints on a probability function. To avoid this inconsistency, the defender of (Ev2) must say that, in fact, our lack of evidence concerning $C$, $D$, $C\ \&\ D$ and $C\ \&\ \neg D$ indeed counts as no evidence concerning $C$ and $D$, but does count as evidence concerning $C\ \&\ D$ and $C\ \&\ \neg D$. How might they do that? Well, they might note that, while $C$ and $D$ are each true in half the possible worlds, since they are atomic, $C\ \&\ D$ and $C\ \&\ \neg D$ are true only in a quarter of the possible worlds. And thus a lack of evidence is in fact evidence against them. But of course this line of argument appeals to the Principle of Indifference. Only if you think that every world should receive equal credence will you think that a lack of evidence counts as no evidence for a proposition that is true at half of the possible worlds, but counts as genuine evidence against a proposition that is true at only a quarter of the worlds.

Thus, I conclude that the HLWW argument fails. While (Ev1) and (Ev2) may be true, we cannot appeal to them in order to justify the Principle of Indifference, since they can only be defended by appealing to the Principle of Indifference itself.

Tuesday, 17 January 2017

Recently, James Hawthorne, Jürgen Landes, Christian Wallmann, and Jon Williamson published a paper in the British Journal of Philosophy of Science in which they claim that the Principal Principle entails the Principle of Indifference -- indeed, the paper is called 'The Principal Principle implies the Principle of Indifference'. In this post, I argue that it does not.

All Bayesian epistemologists agree on two claims. The first, which we might call Precise Credences, says that an agent's doxastic state at a given time $t$ in her epistemic life can be represented by a single credence function $P_t$, which assigns to each proposition $A$ about which she has an opinion a precise numerical value $P_t(A)$ that is at least 0 and at most 1. $P_t(A)$ is the agent's credence in $A$ at $t$. It measures how strongly she believes $A$ at $t$, or how confident she is at $t$ that $A$ is true. The second point of agreement, which is typically known as Probabilism, says that an agent's credence function at a given time should be a probability function: that is, for all times $t$, $P_t(\top) = 1$ for any tautology $\top$, $P_t(\bot) = 0$ for any contradiction $\bot$, and $P_t(A \vee B) = P_t(A) + P_t(B) - P_t(AB)$ for any propositions $A$ and $B$.

So Precise Credences and Probabilism form the core of Bayesian epistemology. But, beyond these two norms, there is little agreement between its adherents. Bayesian epistemologists disagree along (at least) two dimensions. First, they disagree about the correct norms concerning updating on evidence learned with certainty --- some say they are diachronic norms concerning how an agent should in fact update; others say that there are only synchronic norms concerning how an agent should plan to update; and others think there are no norms concerning updating at all. Second, they disagree about the stringency of the synchronic norms that don't concern updating. Our concern here is with the latter. Some candidates norms of this sort: the Principal Principle, which says how an agent's credences in propositions concerning the objective chances should relate to her credences in other propositions (Lewis 1980); the Reflection Principle, which says how an agent's current credences in propositions concerning her future credences should relate to her current credences in other propositions (van Fraassen 1984, Briggs 2009); and the Principle of Indifference, which says, roughly, that an agent with no evidence should divide her credences equally over all possibilities (Keynes 1921, Carnap 1950, Jaynes 2003, Williamson 2010, Pettigrew 2014). Those we might call Radical Subjective Bayesians adhere to Precise Credences and Probabilism, but reject the Principal Principle, the Reflection Principle, and the Principle of Indifference. Those we might call Moderate Subjective Bayesians adhere to Precise Credences, Probabilism, and the Principal Principle (and also, quite often, the Reflection Principle), but they reject the Principle of Indifference. And the Objective Bayesians accept all of the principles.

In a recent paper, Hawthorne et al. (2015) (henceforth, HLWW) argue that Moderate Subjective Bayesianism is an inconsistent position, because the Principal Principle (and, indeed the Reflection Principle) entails the Principle of Indifference. Thus, it is inconsistent to accept the former and reject the latter. We must either reject the Principal Principle, as the Radical Subjective Bayesian does, or accept it together with the Principle of Indifference, as the Objective Bayesian does.

Notoriously, as Lewis originally stated it, the Principal Principle includes an admissibility condition (266-7, Lewis 1980). Equally notoriously, Lewis did not provide a precise account of this condition, thereby leaving his formulation of the principle similarly imprecise. HLWW do not give a precise account either. But they do appeal to two principles that they take to follow intuitively from the Principal Principle. And from these two principles, together with the Principal Principle itself, they derive what they take to be an instance of the Principle of Indifference. The first principle to which they appeal --- their Condition 1 --- is in fact provable, as they note. The second --- their Condition 2 --- is not. Indeed, as we will see, on the correct understanding of admissibility, it is false. Thus, the HLWW argument fails. What's more, its conclusion is not true. It is possible to satisfy the Principal Principle without satisfying the Principle of Indifference, as we will see below. Moderate Subjective Bayesianism is a coherent position.

Introducing the Principal Principle

We begin by introducing the Principal Principle. To aid our statement, let me introduce a piece of notation. Given a proposition $A$ and a real number $0 \leq x \leq 1$, let $C^A_x$ be the following proposition: The current objective chance of $A$ is $x$. And we will let $P_0$ be the credence function of our agent at the very beginning of her epistemic life --- when she is, as Lewis would say, a superbaby; that is, she is not yet in receipt of any evidence. Then, as Lewis originally formulates the Principal Principle, it says this:

In this version, the principle applies to an agent only at the beginning of her epistemic life; it governs her initial credence function. In this situation, the principle says, her credence in a proposition $A$ conditional on the conjunction of some proposition $E$ and a chance proposition that says that the chance of $A$ is $x$ should be $x$, providing the conditional probability is well-defined and $E$ is admissible for $A$.

The motivation for the admissibility condition is this. Suppose $E$ entails $A$. Then we surely don't want to demand that $P_0(A | C^A_xE) = x$. After all, if $x < 1$, then such a demand would conflict with Probabilism, since it is a consequence of Probabilism that, if $E$ entails $A$, then $P_0(A | C^A_xE) = 1$. Thus, we must at least restrict the Principal Principle so that it does not apply when $E$ entails $A$. But there are other cases in which the Principal Principle should not be imposed, even if such an application would not be outright inconsistent with other norms such as Probabilism. For instance, suppose that $E$ entails that the chance of $A$ at some time in the future is $x' \neq x$. Then, again, we don't want to require that $P_0(A | C^A_xE) = x$. The moral is this: if $E$ contains information about $A$ that overrides the information that the current chance of $A$ gives about $A$, then it is inadmissible. Clearly any proposition that logically entails $A$ provides information that overrides the current chance information about $A$; and so does a proposition that entails something about the future chance of $A$. So much for propositions that are inadmissible. Are there any we can be sure are admissible? According to Lewis, there are, namely, propositions solely concerning the past or the present. Thus, Lewis does not give a precise account of admissibility: he gives a heuristic --- $E$ is admissible for $A$ if $E$ does not provide information about $A$ that overrides the information contained in propositions about the current chance of $A$ --- and he gives examples of propositions that do and do not provide such information --- I've recalled some of Lewis' examples here.

Now, as Lewis himself noted, the Principal Principle has implausible consequences when the chances are self-undermining --- that is, when the chances assign a positive probability to outcomes in which the chances are different. This happens, for instance, for Lewis' own favoured account of chance, the Humean account or Best System Analysis. This lead to reformulations of the Principal Principle, such as Thau's and Hall's New Principle (Lewis 1994, Thau 1994, Hall 1994) and Ismael's General Recipe (Ismael 2008). HLWW say nothing explicitly about whether or not chances are self-undermining. But, since they are interested in investigating the Principal Principle and not the New Principle or the General Recipe, I take them to assume that chances are not self-undermining. I will do likewise.

The HLWW argument

However imprecise Lewis' account of admissibility is, HLWW take it to be precise enough to allow us to be confident of the following principles:

Condition 1 If
(1a) $E$ is admissible for $A$, and
(1b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,
then
(1c) $EF$ is admissible for $A$.

Condition 2 If
(2a) $E$ is admissible for $A$, and
(2b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,
then
(2c) $E(A \leftrightarrow F)$ is admissible for $A$.

This is not provable. Indeed, as we will see below, it is false. Nonetheless, together with Lewis' Principal Principle, Conditions 1 and 2 entail a constraint on an agent's credence function that HLWW take to be the constraint imposed by the Principle of Indifference.

Proposition 1 Suppose Lewis' Principal Principle together with Conditions 1 and 2 hold. And suppose that there are propositions $A$, $E$, and $F$ and $0 < x < 1$ such that $E$ is admissible for $A$. Suppose further that $F$ is atomic and contingent. Then

(i) If $C^A_xE$ contains no information that renders $F$ relevant to $A$, then the following is required of the agent's initial credence function: $P_0(F | C^A_xE) = 0.5.$

(ii) If $C^A_xE$ contains no information whatsoever about $F$ (so that $P_0(F | C^A_xE) = P_0(F)$), then the following is required of the agent's initial credence function: $P_0(F) = 0.5$

HLWW take Proposition 1 to show that the Principle of Indifference follows from the Principal Principle. After all, Condition 1 is simply a theorem. And they take Condition 2 to be a consequence of the Principal Principle, given the correct understanding of admissibility. So if you assume the Principal Principle, you get all of the hypotheses of the theorem. However, as we will see in the next two sections, Condition 2 is in fact false.

Now suppose that, if $ch$ is a probability function defined over all the propositions about which the agent has an opinion, $C_{ch}$ is the proposition that says that the objective chances are given by $ch$. Then it follows from the Principal Principle and Current Chance Admissibility that $P_0(A | C_{ch}) = ch(A)$. But it also follows from this that:

This is a version of the Principal Principle that makes no mention of admissibility. From it, something close to Lewis' Principal Principle follows: If $P_0(C^{A|E}_x E) > 0$, then $$P_0(A | C^{A|E}_x E) = x$$ where $C^{A|E}_x$ is the proposition: The current objective chance of $A$ conditional on $E$ is $x$. What's more, while Levi's version does not mention admissibility, since it applies equally when the proposition $E$ is not admissible, it does suggest a precise account of admissibility. And it is possible to show that, if we take the version of Lewis' Principal Principle that results from understanding admissibility in this way, it is a consequence of Levi's Principal Principle.

That is, on this account $A$ is admissible for $E$ if every chance function renders $A$ and $E$ stochastically independent. Three points are worthy of note:

All propositions providing future information about the chance of $A$ or information about the truth value of $A$ are Levi-inadmissible, since $A$ will be stochastically dependent on such propositions according to all possible current chance functions. So this account of admissibility agrees with the examples of clearly inadmissible propositions that we gave above.

All propositions solely about the past are Levi-admissible, since all such propositions will now be true or false and will be assigned chance 1 or 0 accordingly by all possible current chance functions. So this account of admissibility agrees with the examples of clearly admissible propositions that we gave above.

If $A$ is Levi-admissible for $E$, then $P_0(A | C^A_xE) = P_0(A | C^{A|E}_xE ) = x$. That is, Lewis' Principal Principle follows from Levi's version if we understand Lewis' notion of admissibility as Levi-admissibility.

Taken together, (1), (2), and (3) entail that Levi-admissibility has all of the features that Lewis wished admissibility to have.

Now, although Levi's account of admissibility recovers Lewis' examples, it might seem to be too demanding. Suppose, for instance, that $A$ is a proposition concerning the toss of a coin in Quito --- it says that it will lands heads --- while $E$ is a proposition concerning tomorrow's weather in Addis Ababa --- it says that it will rain. Then, intuitively, $E$ is admissible for $A$. But $E$ is not Levi-admissible for $A$. After all, we are considering an agent at the beginning of her epistemic life. And so there are certainly possible chance functions --- probability functions that, for all she knows, give the objective chances --- that do not render $E$ and $A$ stochastically independent.

However, in fact, on closer inspection, the Levi-admissibility verdict is exactly right. Consider my credence in $A$ conditional on $E$ and the chance hypothesis $C^A_{0.5}$, which says that the coin in Quito is fair and so the unconditional chance of $A$ is 0.5. Amongst the chance functions that are epistemically possible for me, some make $E$ irrelevant to $A$, some make it positively relevant to $A$ and some make it negatively relevant to $A$. Indeed, we might suppose that the possible chances of $A$ conditional on $E$ run the full gamut of values from 0 to 1. In that case, surely we don't want to say that $E$ is admissible for $A$ and thereby impose, via the Principal Principle, the demand that our agent's credence in $A$ conditional on $E$ and $C^A_{0.5}$ is 0.5. After all, if I choose to place most of my prior credence on the chance hypotheses on which $E$ is positively relevant to $A$, then my credence in $A$ conditional on $E$ and $C^A_{0.5}$ should not be 0.5 --- it should be something greater than 0.5. If I choose to place most of my prior credence on the chance hypotheses on which $E$ is negatively relevant to $A$, then my credence in $A$ conditional on $E$ and $C^A_{0.5}$ should not be 0.5 --- it should be something less than 0.5. Of course, we might think that it is irrational for our agent, a superbaby with no evidence one way or the other, to favour the positive relevance hypotheses over those that posit neutral relevance and negative relevance. We might think that she should spread her credences equally over all of the possibilities, in which case their effects will cancel out, and her credence in $A$ conditional on $E$ and $C^A_{0.5}$ will indeed be 0.5. But of course to do this is to assume the Principle of Indifference and beg the question.

The failure of Condition 2

With this precise account of admissibility in hand, we can now test to see whether or not it vindicates Condition 2 --- recall, HLWW claim that this is a consequence of the Principal Principle. As we saw above, Condition 2 runs as follows:

Condition 2 If
(2a) $E$ is admissible for $A$, and
(2b) $C^A_xE$ contains no information that renders $F$ relevant to $A$,
then
(2c) $E(A \leftrightarrow F)$ is admissible for $A$.

Now suppose that Lewis' Principal Principle is true, and assume that admissibility means Levi-admissibility. Then this is equivalent to:

Proposition2 For any value $0 \leq y \leq 1$, there is a chance function $ch$ such that (2a$^*$) and (2b$^*$) hold, but $$ch(A | E(A \leftrightarrow F)) = y$$

Thus, (2a$^*$) and (2b$^*$) impose no constraints whatsoever on the chance of $A$ conditional on $E(A \leftrightarrow F)$.

Thus, it is possible that $E$ is Levi-admissible for $A$ and that $C^A_xE$ carries no information whatsoever about $F$, and yet $E(A \leftrightarrow F)$ is not Levi-admissible for $A$. Thus, Condition 2 is false and the HLWW argument fails.

Levi's Principal Principle and the Principle of Indifference

Of course, the failure of an argument does not entail the falsity of its conclusion. It might yet be the case that the Principal Principle entails the Principle of Indifference, even if the HLWW argument does not show that. But in fact we can show that this is not true. To see this, we note a sufficient condition for satisfying Levi's Principal Principle:

Proposition 3 Suppose $C$ is the set of all possible chance functions. Then, if $P_0$ is in the convex hull of $C$, then $P_0(A | C_{ch} E) = ch(A | E)$.

Now, if Levi's Principal Principle entails the Principle of Indifference, and the Principle of Indifference entails that every atomic proposition has probability 0.5, then it follows that every member of the convex hull of the set of possible chance functions must assign probability 0.5 to every atomic proposition. But it is easy to see that this is not true. Let $F$ be the atomic proposition that says that a sample of uranium will decay at some point in the next hour. In the absence of evidence, the possible chances of $F$ range over the full unit interval from 0 to 1. Thus, there are members of the convex hull of the set of possible chance functions that assign probabilities other than 0.5 to $F$. And, by Proposition 3, these members will satisfy Levi's Principal Principle.

Applying Levi's Principal Principle

A possible objection: Levi's Principal Principle is all well and good in theory, but it is not applicable. Suppose we are interested in a proposition $A$; and we have collected evidence $E$. How might we apply Levi's Principal Principle in order to set our credence in $A$? In the case of Lewis' version of the principle, we need only know the chance of $A$ and the fact that $E$ is admissible for $A$, and we often know both of these. But, in order to apply Levi's version, we must know the chance of $A$ conditional on our evidence $E$. And, at least for large and varied bodies of evidence, we never know this. Or so the objection goes.

But the objection fails. In fact, Levi's Principal Principle may be applied in those cases. You don't have to know the chance of $A$ conditional on $E$ in order to set your credence in $A$ when you have evidence $E$. You simply have to have opinions about the different possible values that that conditional chance might take. You then apply Levi's Principal Principle, together with the Law of Total Probability, which jointly entail that your credence in $A$ given $E$ should be your expectation of the chance of $A$ given $E$. Of course, neither Levi's Principal Principle nor the Law of Total Probability will tell you how to set your credences in the different possible values that the conditional chance of $A$ given $E$ might take. But that's not a problem for the Moderate Subjective Bayesian, who doesn't expect her evidence to pin down a unique credal response. Only the Objective Bayesian would expect that. You pick your probability distribution over those possible conditional chance values and Levi's Principal Principle does the rest via the Law of Total Probability.

Conclusion

The HLWW argument purports to show that the Principal Principle entails the Principle of Indifference. But it fails because, on the correct understanding of admissibility, Condition 2 is not a consequence of the Principal Principle; and indeed it is false. What's more, we can see that there are credence functions that satisfy the correct version of the Principal Principle --- namely, Levi's Principal Principle --- that do not satisfy the Principle of Indifference. The logical space is therefore safe once again for Moderate Subjective Bayesians, that is, those who accept Precise Credences, Probabilism, the Principal Principle (and perhaps the Reflection Principle), but who deny the Principle of Indifference.